# College Math Teaching

## September 19, 2014

### Freshman calculus: they don’t always know the basics….

Filed under: basic algebra, calculus — Tags: , — collegemathteaching @ 5:43 pm

Example one: many students don’t know that $\frac{\frac{a}{b}}{c} \ne \frac{a}{\frac{b}{c}}$ (of course, assume that $a, b, c, \ne 0$. ) Where this came up: when we computed $lim_{h \rightarrow 0} \frac{\frac{1}{1+h} - 1}{h}$ we obtained $lim_{h \rightarrow 0} \frac{\frac{-h}{1+h}}{h}$ and a student didn’t understand why this was equal to $lim_{h \rightarrow 0} \frac{-1}{1+h}$

I ended up asking the student to simplify $\frac{\frac{2}{3}}{2} =$ and ….asking: ok, “if I have 2/3’rds of a pie and I give two people an equal piece of that, how much pie does each person get?

Example two: I gave “find the domain of $\frac{1}{\sqrt{x^2 - 9}}$ and two students didn’t understand why an answer of $-3 > x > 3$ was logically impossible. One of them told me: “my calculus teacher in high school told me to do it this way”: I am 99.99 percent that this isn’t true, but, well, I stayed with it until the student understood why such a statement was logically impossible. Oh yes, this same “I had calculus in high school” student was sure that I was wrong when I told him that “the derivative of a constant function is zero”; he was SURE that it is “1”.